Source: PHYS 116 Final Exam Notes, University of Washington
Difficulty: Introductory | Prerequisites: Basic wave properties (wavelength, amplitude, frequency, wave speed)
Big Picture
This is the starting point for understanding what happens when two or more waves meet in the same medium. Before this, you studied individual waves in isolation. Now you need to know how they combine, because nearly every real wave scenario (sound in a room, light through a lens, vibrations on a guitar string) involves multiple waves overlapping. The principle of superposition is the single rule that governs all of it, and the two outcomes, constructive and destructive interference, show up repeatedly through the rest of the course.
When two waves occupy the same space, their displacements add together point by point. This is the principle of superposition. If the waves reinforce each other, the combined amplitude is larger (constructive interference); if they oppose each other, the combined amplitude is smaller or zero (destructive interference). After overlapping, the waves pass through each other and continue unchanged.
Superposition (principle of superposition)
When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave. In simple terms, waves just add together wherever they overlap.
Interference
The superposition of two or more waves. This is the general term for what happens when waves combine, whether the result is larger, smaller, or the same.
Constructive interference
Occurs when two waves overlap and the total displacement is larger than either wave alone. Think of it as the waves teaming up: crest meets crest, so the result is a bigger crest.
Destructive interference
Occurs when two waves overlap and the total displacement is smaller than either wave alone (and can be zero). Think of it as the waves cancelling: crest meets trough, so they reduce or eliminate each other.
Displacement
The distance a point in the medium has moved from its equilibrium (rest) position. In superposition problems, you add displacements algebraically (with sign).
Waves can pass directly through each other. Two sound waves or string pulses travelling in opposite directions do not bounce off one another; they overlap, combine, and then continue on as if nothing happened.
While they overlap, the net displacement at every point equals the algebraic sum of the individual displacements. Positive displacement (crest) plus positive displacement (crest) gives a larger positive value. Positive plus negative (crest plus trough) gives a smaller value, or zero if they are equal and opposite.
After the overlap region, both waves emerge with their original shape and amplitude unchanged.
Occurs when the waves are aligned so that their crests (or troughs) coincide.
The combined displacement in the overlap region is larger than either wave on its own.
Maximum constructive interference happens when two identical waves are perfectly in phase: the resulting amplitude is double the original.
Occurs when a crest of one wave meets a trough of the other.
The combined displacement in the overlap region is smaller than either wave alone.
Perfect destructive interference happens when two identical waves are exactly out of phase: the net displacement is zero at every point during full overlap.
Energy is not lost during destructive interference. The waves' energy is redistributed, largely into kinetic energy of the medium at that instant.
A common point of confusion: when waves cancel, the energy does not vanish. At the instant of perfect cancellation, the displacement is zero but the medium is still moving. The energy sits in kinetic form rather than potential form.
Once the waves pass through each other, all original energy reappears in the separated waves.
The only formula for this section is the superposition principle itself:
y_{\text{net}}(x, t) = y_1(x, t) + y_2(x, t) + \cdotsThe net displacement at any position x and time t equals the sum of each individual wave's displacement at that same position and time. There is no special formula to memorise beyond this additive rule.
Noise-cancelling headphones work by generating a sound wave that is the inverse of ambient noise, producing destructive interference at your ear. Concert halls and recording studios are shaped to manage constructive and destructive interference so that sound reaches listeners evenly rather than creating dead spots or painfully loud zones.
Students often think waves "bounce off" each other when they meet. They do not. Waves pass straight through one another and emerge unchanged.
Students sometimes believe energy is destroyed during destructive interference. Energy is conserved; it is temporarily stored as kinetic energy in the medium while the displacement is zero.
Some students assume constructive interference always doubles the amplitude. It only doubles when the two waves have identical amplitude and are perfectly in phase. Unequal amplitudes or partial phase alignment produce a result somewhere between full constructive and full destructive.
A common error is forgetting to use algebraic (signed) addition. A positive displacement of +3 cm combined with a negative displacement of -2 cm gives +1 cm, not 5 cm.
⚠️ You will almost certainly be asked to sketch or calculate the net displacement when two pulse shapes overlap at a given instant. Practice adding displacements point by point.
⚠️ Expect a question testing whether you know waves pass through each other unchanged after overlapping.
⚠️ Be ready to explain where the energy goes during destructive interference (kinetic energy, not "lost").
True or false: When two waves destructively interfere, energy is destroyed. (False, energy is conserved.)
Fill in the blank: The principle of superposition states that the net displacement equals the ______ of the individual displacements. (sum)
True or false: After two pulses pass through each other, their shapes and amplitudes are changed. (False, they emerge unchanged.)
Fill in the blank: When two identical waves are perfectly in phase, the resulting amplitude is ______ the original. (double / twice)
True or false: Constructive interference occurs when a crest meets a trough. (False, that is destructive interference.)
Q: Two identical pulses travel toward each other on a string. At the moment they fully overlap, the displacement is zero everywhere. Has the energy of the waves disappeared? Explain.
A: No. The energy has not disappeared. At the instant of complete destructive interference the string has zero displacement, but every point on the string is still moving. The energy is entirely in kinetic form at that moment. Once the pulses pass through each other, the original displacement profiles reappear.
Q: Two wave pulses on a string have amplitudes of +4 cm and -1 cm. What is the displacement at a point where both pulses overlap?
A: +3 cm. Superposition is algebraic addition: (+4) + (-1) = +3 cm.
Q: A crest of amplitude A from wave 1 meets a crest of amplitude A from wave 2. What is the resulting amplitude at the overlap, and what type of interference is this?
A: The resulting amplitude is 2A. This is constructive interference.
Q: After two pulses on a string pass through each other, how do their individual shapes compare to what they were before overlapping?
A: They are identical. Waves pass through one another without any permanent change to shape or amplitude.
Superposition is the foundation for standing waves (Ch. 16.2-16.3), where two travelling waves moving in opposite directions combine to create a pattern that appears stationary. It also underpins the interference patterns from two sources (Ch. 16.6), such as two speakers emitting the same tone, and the phenomenon of beats (Ch. 16.7), where two slightly different frequencies produce a pulsing loudness pattern. Every wave-combination topic in the rest of this course relies on superposition.
Superposition, principle of superposition, wave interference, constructive interference, destructive interference, wave addition, algebraic sum of displacements, crest meets crest, crest meets trough, wave overlap, wave energy conservation, noise cancellation, PHYS 116, general physics waves, University of Washington